The orthogonal frequency-division multiplexing method (OFDM) has established itself in many areas for high-rate data transmission in radio systems with large bandwidth. It is used, for example, in digital audio broadcasting (DAB), digital video broadcasting—terrestrial (DVB-T), in wireless local area networks (WLAN) and 4G long term evolution LTE. The principle of OFDM consists of splitting the high-rate overall data stream into many low-rate data streams and transmitting them parallel via the corresponding number of orthogonal subcarriers. Compared to single-carrier methods, various advantages result for transmission via frequency selective multipath channels. In doing so, channel equalization can be implemented efficiently in the frequency range without using costly adaptive equalization filters. Furthermore, by introducing a guard interval (GI) between the OFDM symbols, inter-symbol interference (ISI) can be effectively prevented. Signal generation can be realized with the help of inverse discrete Fourier transform (IDFT) or its expense-favorable implementation, the inverse fast Fourier transform (IFFT).
An almost identical method is also used for line-tied transmission. It is known as “discrete multitone transmission” (DMT). It is used, for example, for broadband digital data transmission via subscriber connection lines (DSL). Because DMT can also be viewed as a form of OFDM, the term “OFDM” will be used for both module types, hereafter. Thus, the following remarks similarly refer to DMT systems.
One problem with the transmission via channels with distinct multi-path dispersion (e.g. by means of signals reflected at buildings during radio transmission or reflection onto lines during line-tied transmission) is that destructive interference can cause erasing of subcarriers (“spectral zeros”) in the channel). Because the symbols modulated onto the affected subcarriers cannot be detected correctly by the receiver, any longer, this causes a high symbol error rate (SER) or a high bit error rate (BER) in the data stream, which only slightly decreases with increasing signal to noise ratio (SNR). Thus, it cannot readily be compensated for by increasing the transmission capacity.
As a matter of principle, the problem can be avoided if the transmitter only modulates the data to the strong subcarriers and omits the weak or deleted subcarriers. This involves, however, that it knows the actual radio channel to the receiver (channel state information at transmitter's), which is not the case in many systems. The reasons for this are, for example, the higher complexity of the system connected therewith, and feedback information that may be used (transmission of channel information from receiver to transmitter), which cause a transmission overhead and have to be made available to transmitter with very little delay for a fast time-variable channel.
Generally, OFDM is coupled with forward error correction (FEC), which results in coded OFDM (COFDM). To do this, targeted redundancy is added to the sent data stream by means of suitable channel coding and used at receiver's to correct transmission errors. By means of FEC, BER can be significantly reduced, whereby the net data rate is decreased by the added redundancy—a defaulted net data rate entails correspondingly increased transmission resources. A linked FEC is oftentimes implemented in classic COFDM systems, where a convolution code is linked as inner code and a block code (e.g. Reed Solomon Code) is linked as outer code, for example. The inner code has the primary task to reduce the effects of the weak subcarriers or the spectral zeros. With the help of the outer code, the error rate is further reduced by several decimal powers.
One alternative to compensate for the influence of weak subcarriers is to spread data symbols prior to OFDM modulation in the frequency range. Every symbol is then no longer transmitted via a single but via all subcarriers, via which the spreading operation is being implemented. Any sub-carrier carries a linear combination of all transmission symbols within the block. Even in case of loss of several subcarriers, it is often times still possible to reconstruct the transmission symbols, and the BER strongly decreases around the operating point (as in the non-frequency selective case) with increasing SNR. This is referred to as diversity gain. The method is referred to as “code-spread OFDM” (CS-OFDM; see reference [1]—the stated references can be found in detail in Appendix 1 to this description), “spread OFDM” (SOFDM; see reference [2]) or “frequency domain spreading” (see reference [3]). The abbreviation CS-OFDM will be used, hereafter. Classically, M modulation symbols are spread onto/over N subcarriers, causing no loss in data rate. If M<N modulation symbols are spread onto N subcarriers, the system is referred to as “partially loaded” (PL) (PL CS-OFDM; see reference [1]). By means of partial load, further gains can be achieved at the expense of the data rate.
Compared to COFDM systems, no redundancy has to be added to CS-OFDM systems to utilize frequency diversity. An additional FEC with the help of channel coding is, however, generally reasonable to further reduce the error rate. Because the channel code used in such case, does not have to compensate for the influence of the weak subcarriers any longer, the code rate can turn out larger for classic coding methods in connection with the spreading.
CS-OFDM signals have unfavorable signal statistics, in common with OFDM signals. Interference of a plurality of subcarriers results in a poor ratio of instant performance to mean signal performance (signal statistics). This is often times indicated by the peak-to-average power ratio (PAPR), the ratio of maximum instant performance to mean signal performance or the crest factor (ratio of the maximum instant value to the root mean square value of the signal). The measurements take on high values for OFDM signals. Signal statistics deteriorates with increasing subcarrier number N.
High requirements to the linearity of the components used in the system, and especially to the power amplifiers, result from poor signal statistics. Therefore, such components can often times only be operated in an inefficient operating point. One problem for the PL CS-OFDM is the fact that the PAPR—in comparison to the fully loaded system—can drastically continue to deteriorate.
The presentation of the problem in respect to a CS-OFDM system is the realization of an efficient, adaptive transmission by means of an implementation with low complexity:                high performance ability: good utilization of the frequency diversity of the frequency selective channel, low bit error rate with given SNR and given net data rate        good signal statistics (low PAPR): low requirements to the linearity of the analog hardware (above all at the power amplifiers) and performance specific operating point        high adaptivity: adaptivity possibility of the transmission (robustness, net data rate, redundancy) to the conditions of the transmission channel with low complexity and fine tuning        implementation with low complexity: low utilization of resources of digital signal processing and low requirements to the speed of digital signal processing, cost and performance efficient implementation        
Generally, the criteria cannot be optimized independently from another. Conventional technology knows several spreading methods with good properties, which will be described, hereafter.
Conventional technology describes Hadamard (also termed “Walsh-Hadamard” or “Hadamard-Walsh”; see references [1]-[4] and [5]), DFT (see references [3], [5]) as well as Vandermonde spreading (see reference [4]). These spreading methods are characterized by high performance ability. This applies at least to large spreading lengths, that is, if the spreading is being implemented by means of a high number of subcarriers. Modifications of the Hadamard transform and the discrete Fourier transform (DFT), the “rotated” Hadamard transform and the “rotated” DFT are suggested in reference [5] to improve performance ability for low spreading lengths.
An increase of CS-OFDM is introduced in references [1], [4]. It consists of feeding a low number of modulation symbols into the spreading operation, which are available as subcarriers, that is, in spreading M symbols onto N subcarriers, where M<N. The system is then referred to as “partially loaded” (PL CS-OFDM). The classic case, a fully loaded system, exists for M=N. Reference [1] suggests the use of PL CS-OFDM to realize further gains compared to OFDM and CS-OFDM, at the expense of the data rate.
For DFT spreading, recourse can be made to the fast Fourier transform (FFT), that is, an efficient algorithm for calculation of the discrete Fourier transform (DFT). The complexity amounts to O (N log2 (N)), with multiplications with the root of unity and additions being performed. The Vandermonde spreading, a multiplication of a Vandermonde matrix of the dimension N×N with a symbol vector of the length N, can be realized with the complexity O (N log2 (N)) (multiplications and additions; see reference [17]). Hadamard spreading can be implemented with the help of the fast Walsh-Hadamard transform (FWHT). It involves N log (N) additions or subtractions and thus has the lowest computational complexity among the methods with full flexibility (fine-tuned partial load, selection of the carriers).
An implementation of the DFT spreading with particularly low complexity is described in reference [16]. In doing so, the multiple carrier system is reduced back to a single carrier system, so that the transmitter can be realized very easily, which, however, is not similarly applicable to the receiver in case a channel estimation and a channel equalization have to be implemented. Furthermore, the system is no longer fully flexible with respect to the partial load with full diversity and selection of the allocated subcarriers.
FIG. 1 shows a block diagram by means of which signal processing is explained in a conventional CS-OFDM transmitter. By means of an introduction, it should be noted that the mathematical notation used in this description is explained in Appendix 2 to this description. The matrices and sequences, to which reference is being made, are explained, there, as well. For better understanding, cross references to the sections of Appendix 2 are indicated at several locations. The abbreviations used are listed in Appendix 3 to this description.
FIG. 1 shows a processing chain 100 for generation of the spread OFDM signal. Mathematically, this can be illustrated with the help of vectors, matrices and corresponding operations. The stream of the (already coded, where applicable) data symbols ds is transformed with the help of a serial/parallel converter 102 in data vectors d of the length M. Spreading is being implemented at block 104. The type of spreading is entirely described by the spreading matrix D. Subsequently, the spread data vectors x are fed to IDFT 106. Blocks 104 and 106 can be combined in one block 108, which implements both the spreading and the IDFT. The combination of both operations is hereafter referred to as “overall transform”, and block 108, which implements such transform, as “overall transformer”. The transforming output vectors w can then be submitted to further operations in the digital baseband (e.g. inserting a Guard interval, fenestration). By means of a digital/analog converter, they are converted to analog signals and run through the typical processing chain of a transmitter for digital data processing, until emission via the antenna(s).
Mathematically, the spread data vector x results from a vector matrix multiplication of the spreading matrix D (dimension N×M) with the input vector d (length M, column vector):x=Dd. 
IDFT 106 can be illustrated as multiplication with the IDFT matrix F−1 (see section 2.10 of Appendix 2) of the dimension N×N. We have:w=F−1x and consequentlyw=F−1Dd=:Bd, with the multiplication of vector d with the matrix B=F−1D illustrating the overall transform 108.
The spreading methods known in the art (see, for example, references [1]-[5]) can be characterized by means of matrices, in particular Hadamard, Vandermonde, and DFT matrices, (see definitions in sections 2.8, 2.9 and 2.10 of Appendix 2). With a full load of the system (M=N), D is identical to the characterizing matrix. With a partial load (M<N), a partial block or sub-block of the characterizing N×N matrix is used for D, which block consists of the first M matrix columns (see references [1], [4]). Spreading methods, which are based on a spreading matrix, are referred to as matrix-based spreading methods, hereafter.